Rescaling and shifting

Transcription

1 Rescaling and shifting A fancy way of changing one variable to another Main concepts involve: Adding or subtracting a number (shifting) Multiplying or dividing by a number (rescaling)

2 Where have you seen this before? Going from Fahrenheit to Celsius C = (5/9)*(F-32) Going from Celsius to Fahrenheit F = [(9/5)*C]+32 Going from pounds to kilograms 1 lb = kg Going from kilograms to pounds 1 kg = lbs

3 What does adding a constant do to data? All measures of position (5 number summary, mean) will increase (if adding) or decrease (if subtracting) by the constant All measures of spread (range, IQR, standard deviation) STAY THE SAME

4 Example Say we have the following temperatures (in Fahrenheit): 32, 34, 33, 36, 38, 38, 21 5 number summary: Min: 21 Q1: 32 Median: 34 Q3: 38 Max: 38 IQR= 6 s = 5.84

5 Example (con t) Now say we subtract 32 from each data value Temperatures become: 0,2,1,4,6,6,8 5 number summary: Min: -11 Q1: 0 Median: 2 Q3: 6 Max: 6 IQR= 6 s = 5.84

6 Example (con t) Can see comparing the two that IQR and s didn t change by subtracting 32 from each temperature The 5 number summary changed by subtracting 32 from each element Bottom line: shifting data DOES NOT change the spread

7 What does multiplying or dividing by a number do to data? Changes the: position spread If we multiply all the data by a number, measures of position and measures of spread are multiplied by that number If we divide all the data by a number, measures of position and measures of spread are divided by that number

8 Example (con t) Say we multiply the previous temperatures by (5/9) The temperatures of the original data are now in degrees Celsius : 1.11, 0.55, 2.22, 3.33, 3.33, -6.11

9 For the Celsius data: 5 number summary: Min: Q1: 0 Median: 1.11 Q3: 3.33 Max: 3.33 IQR = 3.33 s = Example (con t)

10 Example (con t) We can see both measures of position and measures of spread change All measures of position and spread were multiplied by (5/9) Bottom line: rescaling data DOES change spread

11 Standardizing variables This is just a special application of shifting and rescaling We shift by subtracting the mean We scale by dividing by the standard deviation

12 Standardizing variables z = y y z has no units (just a number) Puts variables on same scale Mean (center) at 0 Standard deviation (spread) of 1 Does not change shape of distribution s

13 Standardizing variables z = # of standard deviations away from mean Negative z number is below mean Positive z number is above mean

14 Why standardize variables? It is a way to find how many standard deviations from the mean something is It is a way to compare and individual value to a data set It is a way to compare two different looking values

15 Standardizing Variables Height of women Height of men x y = y = x 66, s = 70, s = I am 67 inches tall My friend Dirk is 72 inches tall Who is taller (comparatively)?

16 Standardizing Variables y y z = = = 0.4 s y 2.5 x x z = = = 0.67 s x 3

17 Standardizing Variables I am 0.4 standard deviations above mean height for women Dirk is 0.67 standard deviations above mean height for men Dirk is taller (comparatively)

18 SAT vs. ACT You took SAT and scored 550 Your friend took ACT and scored 30 Which score is better? SAT has mean 500 and standard deviation 100 ACT has mean 18 and standard deviation 6

19 SAT vs. ACT Your score Friend s score

20 SAT vs. ACT Your score Friend s score = 0.5

21 SAT vs. ACT Your score Friend s score = = 2 Your friend scored better on ACT than you did on SAT

22 Heights of 150 Stat 101 Women Heights # Heights # 59.5 < X < X < X < X < X < X < X < X < X < X < X < X < X < X

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24 Height of 150 Stat 101 Women Distribution Shape Symmetric Unimodal Bell-Shaped Center around 66.5 Spread from 59.5 to 73.5 Model with a Normal Distribution

25 Normal Distributions Bell Curve Physical Characteristics Ex. Height Ex. Weight Ex. Length of wings of birds Most important distribution in statistics

26 Normal Distributions Two parameters (not calculated) Mean µ (pronounced meeoo ) Locates center of curve Splits curve in half Standard deviation σ (pronounced sigma ) Controls spread of curve Ruler of distribution Write as N(µ,σ)

27 Standard Normal Distribution Puts all normal distributions on same scale z = y µ σ z has center (mean) at 0 z has spread (standard deviation) of 1

28 Standard Normal Distribution z = # of standard deviations away from mean µ Negative z, number is below the mean Positive z, number is above the mean Written as N(0,1)